Abstract

The present paper deals with the wave propagation in a particular two dimensional structure, obtained from a localized perturbation of a reference periodic medium. This reference medium is a ladder like domain, namely a thin periodic structure (the thickness being characterized by a small parameter ) whose limit (as tends to 0) is a periodic graph. The localized perturbation consists in changing the geometry of the reference medium by modifying the thickness of one rung of the ladder. Considering the scalar Helmholtz equation with Neumann boundary conditions in this domain, we wonder whether such a geometrical perturbation is able to produce localized eigenmodes. To address this question, we use a standard approach of asymptotic analysis that consists of three main steps. We first find the formal limit of the eigenvalue problem as the tends to 0. In the present case, it corresponds to an eigenvalue problem for a second order differential operator defined along the periodic graph. Then, we proceed to an explicit calculation of the spectrum of the limit operator. Finally, we prove that the spectrum of the initial operator is close to the spectrum of the limit operator. In particular, we prove the existence of localized modes provided that the geometrical perturbation consists in diminishing the width of one rung of the periodic thin structure. Moreover, in that case, it is possible to create as many eigenvalues as one wants, provided that is small enough. Numerical experiments illustrate the theoretical results.

Abstract

Photonic crystals, also known as electromagnetic bandgap metamaterials,
are 2D or 3D periodic media designed to control the light
propagation. Indeed, the multiple scattering resulting from the periodicity of
the material can give rise to destructive interferences at some range
of frequencies. It follows that there
might exist intervals of frequencies (called gaps) wherein the monochromatic waves
cannot propagate. At the same time, a local perturbation of the
crystal can produce defect mid-gap modes, that is to say
solutions to the homogeneous time-harmonic wave equation, at a fixed frequency located inside one gap,
that remains strongly localized in the vicinity of the
perturbation. This localization phenomenon is of particular interest
for a variety of promising
applications in optics, for instance the design of highly efficient waveguides [26, 27].

From a mathematical point of view, the presence of gaps is theoretically
explained by the band-gap structure of the spectrum of the periodic partial
differential operator associated with the wave propagation in such
materials (see for instance [12, 32]). In turn, the localization effect is directly linked to the
possible presence of discrete spectrum appearing when perturbing the
perfectly periodic operator. A thorough mathematical description of
photonic crystals can be found in [38]. Without being exhaustive, let us
remind the reader about a few important results on the topic.
In the one dimensional case, it is
well-known [7] that a periodic
material has infinitely many gaps unless it is constant. By contrast,
in 2D and 3D, a periodic medium might or might not have
gaps. Nevertheless, several configurations where at least one gap
do exist can be
found in [17, 18, 25, 42, 44, 3, 29, 30] and
references therein. In
any case, except in dimension one, the number of gaps is expected to
be finite. This statement, known as the Bethe Sommerfeld conjecture is
fully demonstrated in
[49, 50] for the periodic Schrödinger operator
but is still partially open for Maxwell equations (see
[58]). For the localization effect, [15, 16, 1, 33, 37, 45, 8, 9]
several papers exhibit situations where a compact (resp. lineic) perturbation of a
periodic medium give rise to localized (resp. guided) modes. It seems that the first results concern strong material perturbations : for local perturbation [15, 16] and for lineic perturbation [33, 37]. There exist fewer results about weak material perturbations : [8, 9] deal with 2D lineic perturbations. Finally, geometrical perturbations are considered in [40, 45], where the geometrical domain under investigation is exactly the same as ours but with homogeneous Dirichlet boundary conditions on the boundary of the ladder.
As in our case, changing the size of one or several rungs of the ladder can create eigenvalues inside a gap (see also remark 7).

The aim of this paper is to complement the references
mentioned above by proving the existence of localized midgap
modes created by a geometrical perturbation of a particular periodic medium. We
shall use a standard approach of analysis (used in
[17, 44]) that consists in
comparing the periodic medium
with a reference one, for which
theoretical results are available. To be more specific, we are interested in the
Laplace operator with Neumann boundary condition
in a ladder-like periodic
waveguide. As the thickness of the rungs (proportional to a small parameter ) tends to zeros, the domain
shrinks to an (infinite) periodic graph. More precisely, the spectrum of the
operator posed on the 2D domain tends to the spectrum of a
self-adjoint operator posed on the limit graph
([54, 39, 55, 51, 48]).
This limit operator consists of the second order derivative operator on
each edge of the graph together with transmission conditions (called
Kirchhoff conditions) at
its vertices ([14, 10, 39]). As
opposed to the initial operator, the spectrum of the limit operator
can be explicitly determined
using a finite difference scheme ([2, 13]). From a mode of the limit operator, we construct a so called quasi-mode and we are able to
prove that, for sufficiently small, the diminution of the thickness of one rung of the ladder gives rise to
a localized mode. Moreover, diminushing , it is possible to create as many eigenvalues as one wants. We point out that the analysis of quantum graphs has been a very
active research area for the last three decades and we refer the reader to
the surveys
[34, 35, 36]
as well as the books [4, 52] for an overview and an exhaustive bibliography of
this field.

In the present work we study the propagation of waves in a ladder-like
periodic medium (see figure a). The homogeneous domain
(we will call it ladder) consists of the
infinite band of height minus an infinite set of equispaced
rectangular obstacles. The domain is -periodic in one space
direction, corresponding to the variable . The distance between two
consecutive obstacles is equal to the distance between the obstacles and
the boundary of the band and is denoted by .

(a) The
unperturbed periodic ladder

(b) The
perturbed ladder

Figure 1: The unperturbed and the perturbed periodic ladders

Remark 1.

Some extensions
We can change the distance between 2 consecutive obstacles from to . The study will depend on this new parameter but its conclusions remain the same.

The aim of this work is to find localized modes, that is solutions of the homogeneous scalar wave equation with Neumann boundary condition

(1)

which are confined in the -direction.
Without giving a strict mathematical formulation (this will be done in the following section) a localized mode can be understood as a solution of the wave equation (1), which is harmonic in time

(2)

where the function (which does not depend on time) belongs to . The factor shows the harmonic dependence on time. Injecting (2) into (1) leads to the following problem for the function :

(3)

Problem (3) is an eigenvalue problem posed in the
unbounded domain . In order to create
eigenvalues, we introduce a local
perturbation in this perfectly periodic domain (the delicate question of existence of eigenvalues (or flat bands) for the unperturbed problem is not addressed in this paper, see for the absence of flat bands in waveguide problems (or the absolute continuity of the spectrum) for instance [56, 22, 57] and for the existence [19] ). The perturbed domain is obtained by changing the
distance between two consecutive obstacles from to
, (see
Figure b in the case where ). It
corresponds to modify the width of one vertical rung of the ladder
from to .

As we will see such a perturbation does not change the continuous
spectrum of the underlying operator but it can introduce a non-empty
discrete spectrum. Our aim is to find eigenvalues by playing with the
values of
and , being treated as a small parameter.

Remark 2.

Imitating the approach developped in this article, it is also possible to study sufficient conditions which ensures the existence of guided modes in a ladder-like open periodic waveguides. More precisely, the domain is minus an infinite set of equispaced perfect conductor rectangular obstacles with Neumann boundary conditions. And this domain is perturbed by a lineic defect, by changing the distance between two consecutive columns of obstacles. There exists a guided mode with a wave number if and only if there exists a localized mode in a perturbed periodic ladder where -boundary conditions are imposed. The results of the present paper can be extended and the sufficient condition which ensures the existence of guided modes remains basically the same.

This section describes a mathematical framework for the analysis of the
spectral problem formulated above. We introduce the
operator , acting in the space
, associated with the eigenvalue
problem (3) in the perturbed domain:

Here .
The operator is self-adjoint and positive. Our goal is to characterize its spectrum and, more precisely, to find sufficient conditions which ensures the existence of eigenvalues.

3.1 The essential spectrum of

To determine the essential spectrum of the
operator , we consider the case ,
where the domain is perfectly periodic (see Figure
a). We will denote the corresponding operator
. The Floquet-Bloch theory shows that the spectrum of periodic elliptic operators is reduced to its essential spectrum and has a band-gap structure [12, 53, 32]:

(4)

where, in the previous formula, the union disappears if .
For , the intervals are called spectral gaps. Their number
is conjectured to be finite (Bethe-Sommerfeld, 1933, [49, 50, 58]). The band-gap structure of the spectrum is a consequence of the following result given by the Floquet-Bloch theory:

(5)

Here is the Laplace operator defined on the
periodicity cell
(see Figure 2) with -quasiperiodic
boundary conditions on the lateral boundaries and
homogeneous Neumann boundary conditions on the remaining part
of the boundary: for ,

Figure 2: Periodicity cell

For each the operator
is self-adjoint, positive and its resolvent is compact. Its spectrum is then a sequence of non-negative eigenvalues of finite multiplicity tending to infinity:

(6)

In (6) the eigenvalues are repeated with their
multiplicity. The representative curves of the functions are called dispersion curves and are known to be continuous and non-constant (cf. Theorem
XIII.86, volume IV in [53]). The fact that the dispersion curves are non-constant implies that the operator has no eigenvalues of infinite multiplicity. Finally, (5) can be rewritten as

which gives (4). The conjecture of
Bethe-Sommerfeld means that for large enough the intervals
overlap or only touch.

Since
and the operators have real coefficients, the function
are even. Thus, it is sufficient to consider
in (5). This will be
used systematically in the rest of the paper.

As expected (this is related to Weyl’s Theorem, see in
[53, Chapter 13, Volume 4], [6, Chapter 9] and
[15, Theorem 1]), the essential spectrum is stable under a perturbation of the thickness
of one rung of the ladder. In the present case, the domains of definition of the resolvents of and are not the same. As a result, we cannot directly apply the standard results (in [6, Chapter 9]).

Proposition 1.

.

This stability result is given in [47, §4 Chapter 3, Theorem 4.1 Chapter 5] and [41, Theorem 5]. For the sake of completeness, we provide a constructive proof based on the following assertion.

Lemma 1.

Let be a function such that

,

such that .

If is a singular sequence for the operator corresponding to the value , then there exists a subsequence of which is also a singular sequence for the operator corresponding to the value .

Proof.

By definition of a singular sequence, the sequence has the following properties:

;

;

in ;

in .

Let us show that there exists a subsequence of which has the same properties. The property 1 is verified by the whole sequence thanks to property (a) . To prove property 2, it suffices to show that there exists a subsequence, still denoted , such that

Then properties 3 and 4 imply that is bounded in . By compactness, one can thus extract a subsequence that converges weakly and strongly in to a limit which is necessarily thanks to property 3, which proves (7). For the sequel, we work with the above subsequence.
The property 3 being obvious the only thing to show is the property 4 for the sequence . We have:

The first and the last terms in the right-hand side tend to zero
thanks to property 4 and (7)). Let us estimate
the second term. Using first property (b), then properties (a) and (1)
together with an integration by parts, we obtain

which tends to due to (7) since is bounded in as well as (by properties 3 and 4).
∎

It is sufficient to take a function in the previous lemma which does not depend on , vanishes in a neighbourhood of the perturbed edge and such that vanishes in a neighbourhood of all vertical edges. Then, it follows from Lemma 1 that any singular sequence associated to of the operator provides the construction of a singular sequence of the operator for the same and vice versa.
∎

The essential spectrum of the operator having a band-gap structure, we will be interested in finding eigenvalues inside gaps (once the existence of gaps is established).

3.2 Towards the existence of eigenvalues: the method of study.

Our analysis consists of three main
steps.

First, we find a formal limit of the eigenvalue problem (3) when
(Section 4.1). To do so, we use the fact that, as
goes to zero, the domain
shrinks to a graph . As a consequence, the
formal limit problem will involve a self-adjoint operator associated with a
second order differential operator along the graph. Its definition
is strongly related to the fact that homogeneous Neumann boundary conditions
are considered in the original problem.
More precisely, at the limit , looking for an eigenvalue
of leads to search an eigenvalue of .
This operator, that is well known (see the works of [14, 10, 39]), will be described
more rigorously in the next section.

The second step is an explicit calculation of the spectrum of
the limit operator. The essential spectrum is determined using the
Floquet-Bloch theory (by solving a set of cell problems) (Section 4.2) while the
discrete spectrum of the perturbed operator is found using a
reduction to a finite difference equation (Section 4.3). In
particular, we shall show that the limit operator has infinitely many
eigenvalues of finite multplicity as
soon as (and no one when ), which form a
discrete subset of .

Finally, when , we deduce the existence of an eigenvalue of close to the eigenvalue of as soon as is small enough (Section 5). The
proof will be based on the construction of a quasi-mode (a kind approximation of the eigenfunction) and a criterion for
the existence of eigenvalues of self-adjoint operators (see for instance Lemma 4 in [43]). It can be seen as a generalization of the well-known min-max principle for eigenvalues located below the lower bound of the
essential spectrum.

An essential preliminary step is the decomposition of the operator
as the sum of two operators, namely its
symmetric and antisymmetric parts. To do so, we introduce the
following decomposition of :

where and are subspaces consisting of functions respectively symmetric and antisymmetric with respect to the axis :

The operator is then decomposed into the orthogonal sum

Accordingly, the limit operator is decomposed as:

(8)

The key point is that, as we shall see, contrary to the full operator
whose spectrum is , both operators and have spectral gaps (an infinity of them), each of them containing eigenvalues : these are isolated eigenvalues for or , but embedded eigenvalues for
. One deduces that the operators
and have at
least finitely many spectral gaps, the number of gaps tending to when goes to 0: this is an important fact for applying the quasi-mode approach.

At this stage, it is worthwhile mentionning that the convergence of the spectrum of differential operators in thin
domains degenerating into a graph is not a new subject, particularly
in the case of elliptic operators. In particular, for the Laplace
operator with Neumann boundary conditions and in the case of compact
domains, the convergence results (which are reduced to the convergence
of eigenvalues) have been known since the works of
Rubinstein-Schatzman [54] and
Kuchment-Zheng [39]. Thanks to the Floquet Bloch
theory, such results are transformed into analogous results for
thin periodic domains (in [39, Theorem 5.1]), since in this case, only continuous spectrum is
involved. For general unbounded domains, a general (and somewhat
abstract) theory has been developed by Post in [51] for
the convergence of all spectral components. This theory can be applied
to our problem, however, for the sake of simplicity, we have chosen to use here a more direct approach (based on the construction quasi-modes).

4.1 The operator .

As , the domain tends
to the periodic graph represented on Figure 3. Let us
number the vertical edges of the graph from left to right so that
the set of the vertical edges is . The upper
end of the edge is denoted by and the lower one by
. The set of all the vertices of the graphs is thenThe horizontal edge joining the
vertices and is denoted by
. The set of all the edges of the graph isand we denote by
the set of all the edges of the graph containing the vertex .Figure 3: Limit graph If is a function defined on we will use the following
notation:Let be a weight
function which is equal to on the ”perturbed edge” , i.e. the limit of the perturbed rung , and to on the other edges:(9)Let us now introduce the following functional spaces(10)(11)where denotes the space of continuous functions on :

We define the limit operator in
as follows. Denoting the restriction of
to ,(12)(13)where stands for the derivative of the function
at the point in the outgoing direction. The vertex
relations in (13) are called Kirchhoff’s
conditions. Note that they all have an identical expression except at the vertices
.
The following assertion as well as its proof can be found in
[34, Section 3.3].

Proposition 2 (Kuchment).

The operator in the space is self-adjoint. The corresponding closed sesquilinear form has the following form:

As for the ladder, we introduce the following decomposition of the space into the spaces of symmetric and antisymmetric functions:Again, the operator can be decomposed into the orthogonal sumwithwhich impliesThus, it is sufficient to study the spectrum of the operators and separately. The analysis of
these two operators being analogous, we will present a detailed study
of (Section 4.2 and
Section 4.3) and state the results for (Section 4.4).

4.2 The essential spectrum of the operator

We shall study the spectrum of the operator by a
perturbation technique with respect to the case which
corresponds to the purely periodic case. The corresponding operator
will be denoted by .
Indeed, based on compact perturbation arguments ( in [6, Theorem 4, Chapter 9]), we
can prove the following proposition:

Proposition 3.

The essential spectra of and
coincide:(14)

This reduces the study of the essential spectrum of to the
study of the spectrum of the purely periodic operator , which can
be done through the Floquet-Bloch theory.

Description of the spectrum of through Floquet-Bloch theory

As previously explained, the spectrum of the operator
can be studied using the Floquet-Bloch theory. One has then to study a
set of problems set on the periodicity cell of . Since
we consider the subspace of symmetric functions with respect to the
axis , this enables to reduce the problem to the lower half part
of the periodicity cell (see Figure 4).Figure 4: Periodicity cellWe introduce the spaces and analogously to (10), (11):We have then(15)where is the following unbounded operator in (16)In the definition of , the condition corresponds to the symmetry with respect to , (16)-(a) is the Kirchhoff’s condition with and
(16)-(b) are the -quasiperiodicity conditions.
For each , the operator is self-adjoint and positive and its resolvent is compact due to the compactness of the embedding . Consequently, its spectrum is a sequence of non-negative eigenvalues of finite multiplicity tending to infinity:(17)In the present case, the eigenvalues can be computed explicitly.

Proposition 4.

For , if and only if is a solution of the equation(18)

Proof.

If is an eigenvalue of the operator then the corresponding eigenfunction is of the form(19)(20)(21)Taking into account that , we arrive at the following linear system(22)(23)(24)(25)(26)The relations (22) express the continuity of the eigenfunction at the vertex . The equation (23) comes from the condition . The relation (24) corresponds to (16)-(a) while (25) and (26) correspond to (16)-(b).
Adding and substracting (25) and (26) lead to
which we can substitute into (22)–(24) to obtain the following system in (27)It is then easy to conclude since one obtains, after some computations omitted hereFor , the relations (19)–(21) are replaced byUsing the fact that we have (instead of (22-26)):One then easily sees that there exists a non-trivial solution if and only if and that the corresponding eigenfunction is constant. Noticing that, for , is solution of (18) allows us to conclude.
∎

The reader will notice that when , the spectrum of has a particular structure: it is the image by the function of a periodic countable subset of . To see that, it suffices to remark that both functions at the left and right hand sides of (18) are periodic with a common period. As a consequence of (15), the spectrum of is the image by the function of a periodic subset of .

Characterization of the spectrum of

Using (15), Proposition 4 allows us to describe the structure of the spectrum of the operator . We first prove the existence of a countable infinity of gaps.

Proposition 5.

The following properties hold, where
and .The operator has infinitely many gaps whose ends tend to infinity.

Proof.

For or , the equation
(18) is satisfied for
so that belongs to .Let (such that ), let us distinguish two cases:

(a) : the left hand side of equation
(18) vanishes for all and, as , the right hand side does not. Then does not belong to the spectrum of . Since and belong to (in view of the point 1) there exists a gap which contains , strictly included in .

(b) : (this case can occur only for special values of , see remark 3), we know by point 1, that and we are going to show that it exists such that
and are in the resolvant set of . This will show the existence of two disjoint gaps of the form and . Setting in relation (18) leads to(28)We havewhich cannot vanish for for small enough,
since . This implies that for all .

The conclusion follows from the fact that the intervals are disjoint, go to infinity with ,
and contain one or two gaps.
∎

Remark 3.

The case 2.(b) of the above proof can occur only for special values of . Indeed, the reader will easily verify that the existence of such that is equivalent to the fact that(29)In fact, the condition (29) also influences the nature of the spectrum of . Indeed it can be shown that when does not belong to , the point spectrum of is empty (i. e. the spectrum of is purely continuous). When belongs to , it coincides with an infinity of eigenvalues of infinite multiplicity, associated with compactly supported eigenfunctions.
It is worth noting that the presence of such eigenvalues is a
specific feature of periodic graphs (see
[35, Section 5]).

Remark 4.

In the
proof, in the case 2.(a), gaps are located in the
vicinity of the points satisfying . These
points are nothing else but the eigenvalues of the 1d Laplace operator
defined on the vertical half edges
with Dirichlet boundary condition at and Neumann boundary
condition at . The presence of gaps is therefore consistent with
[35, Theorem 5] dealing with gaps created by
so-called graph decorations. Indeed, the vertical half edges can be
seen as
decorations of the infinite periodic graph made of the set of the horizontal edges .

Next, we give a more precise description of the gap structure of through a geometrical interpretation of (18).
We first remark that as soon as , (i. e. is solution of (18)) if and only if(30)where the functions and are defined by(31)In the following we reason in the -plane with an additional auxiliary variable. We introduce the domain (32)

Lemma 2.

The domain is the domain of the -plane,
-periodic with respect to , given by(33)

Proof.

The -periodicity of the domain with respect to
follows from the identity . To conclude, it suffices to remark that, for a given , if
varies in the interval is continuous and strictly decreasing from to while, if
varies in the interval , is continuous and strictly decreasing from to
.
∎

Figure 6: The images of the spectral gaps by . In
the left picture, the three types of gaps are distinguished
(according to the legend). Figure 7: An example of eigenvalue of infinite
multiplicity () obtained for . This eigenvalue separates a gap of type (ii) on
the left from a gap of type (iii) on the right. This occurs for
.In other words, is the union of ,
, and the image by the application of the projection on the
line of the intersection of the domain
with the curve . Thanks to this geometrical
characterization, we shall be able to describe the structure of the
gaps of the operator .
Let us introduce the -periodic functions , such that, for any ,

The easy proof of the following result is left to the reader (see
also Figures b, b and 7):

Proposition 6.

An interval is a gap of the operator
if and only if and
one of the following three possibilities holds:

There exists such that , and, ,
.

There exists such that , and ,
.

There exists such that , and ,
.

4.3 The discrete spectrum of

We are now interested in determining the discrete spectrum of
. Suppose that is not in the essential spectrum
of , which implies in particular that (see Prop. 5). Let be a corresponding eigenfunction and let (we consider symmetric functions). Since
the eigenfunction verifies the equation on each
horizontal edge of the graph , one has

(35)

We
first begin by excluding some particular cases:

Lemma 3.

If , then is not in the
discrete spectrum of .

Proof.

If and , then is an eigenvalue of infinite multiplicity (see Remark 3). Thus, it does not belong to the discrete spectrum of . Similarly, if and , then (Prop. 5), which implies that it does not belong to the discrete spectrum of .
∎

Thus we can assume that . In
this case, on the
vertical edges , is given by

(36)

According to (35)-(36), the function is completely
determined by the point values . Moreover, in order to
ensure that , the sequence must
be square integrable:

(37)

It remains to express that
belongs to (see (13)), i.e. Kirchhoff’s conditions
are satisfied.
Doing so, we obtain the following set of finite difference equations:

(38)

(39)

with

(40)

(41)

where is defined in (31).
Thus, we reduced the initial problem for a differential operator on the graph to a problem for a finite difference operator acting on sequences . Looking for particular solutions of (38) for and
under the form leads to
the characteristic equation

(42)

At this point, we observe the following property

Lemma 4.

As soon as , one has the
equivalence

Proof.

Indeed, is equivalent to the existence of
such that

Since , this is equivalent to the
characterization (18) of the essential spectrum.
∎

Since does not belong to the essential spectrum of
, and the discriminant of (42)
is strictly positive, which means that (42)
has two distinct real solutions. Since the product of
these solutions is equal to one, (42) has a
unique solution given by

Remark 5.

Let be a gap of the operator . Since in , is well defined and continuous in . However, might blow up (together with the function ) as tends to or , i.e. at the extremities of the gap.

Theorem 1.

For , the discrete spectrum of the operator is empty. For , let be a gap of the operator :

If is a gap of type (i), then has exactly two simple eigenvalues and that satisfy
.

If is a gap of type (ii) or (iii), then has exactly one simple eigenvalue such that .

Proof.

Assume that . If belongs to the discrete sprectrum of , then is in a gap of and Equation (46) is satisfied. But this is impossible because , which means in particular that .

Then, we consider the case . We investigate the variations of for the different types of gaps described in Prop. 6:

Gap of type (i): as a preliminary step, one can verify that (using for instance the definition (40) of together with the fact that and , see Prop. 6), which implies that

(47)

Then, let us investigate the variations of the function , with : first, since and (see Prop. 6), the strictly decaying function , which is continuous in the interval , has exactly one zero in . We denote it by . Besides, the fonction is continuous and strictly decaying in the interval (Prop. 6 ensuring the existence of such that , we deduce that is continuous in ). Moreover, it satisfies and . Indeed, a direct computation shows that

(48)

As a consequence, and . As a result, has exactly one zero in . We denote it by .

Noting that (47) implies that , we deduce that the function , which is continuous on , is strictly decaying from to in the interval , is strictly increasing from to in the interval , and is negative in the interval .
It follows that , which is therefore also continuous in , is strictly decaying from to in the interval , is negative in the interval , and is strictly increasing from to in . As a result, for any , Equation (46) has exactly two solutions in , the first one belonging to and the second one to .

Gap of type (ii): in this case, and the function blows up in the neighborhood unless . More precisely, we can prove that

By contrast, since and as for the first kind of gap, we can prove that

(49)

Then, here again, we investigate the variations of the function , with . In view of Prop 6, the function is continuous, strictly decaying and negative in the intervall .
Then, the function is continuous in , strictly decaying, and (thanks to (48)) satisfies

As result, has still exactly one zero in . We denote it by .

Noting that (49) implies that , we deduce that the function , which is continuous in , is negative in , and stricly increasing from to in . Thus, the function , which is continuous in , is negative in , and strictly increasing from to on . Consequently, for any , Equation (46) has exactly one solution (that belongs to ). The proof for the gaps of type (iii) follows the same way.

∎

4.4 The spectrum of the operator .

We will now briefly describe the modifications of the previous considerations in the case of the operator .
The operator corresponding to the periodic case is denoted by . First, based on compact perturbation arguments, we can prove the proposition, which is analogous to Proposition 3:

Proposition 7.

The essential spectra of and
coincide:

(50)

Besides, using the Floquet-Bloch Theory, we obtain the analogue of Proposition 4 in the antisymmetric case (we refer the reader to Section 4.2 for the definition of ):

Proposition 8.

For , if and only if and is a solution of the equation

(51)

Thanks to the previous characterization, and similarly to the results of Proposition 5, we can describe the structure of the spectrum of :

Proposition 9.

The following properties hold:

, where
and .

The operator has infinitely many gaps whose ends tend to infinity.

Then, excluding here again the particular cases , the computation of the discrete spectrum leads to the set (38)-(39) of finite-difference equations substituting and for respectively and :

The investigation of the characteristic equation (42) then provides the following characterization for the discrete spectrum of :

(52)

Finally, as in the symmetric case (see Theorem 1), a detailed analysis of (52) allows us to prove the following result of the existence of eigenvalues:

Theorem 2.

For the discrete sprectrum of is empty. For , there exists either one or two eigenvalues in each gap of .

4.5 The spectrum of the operator .

As we have seen, both of the operators and have infinitely many gaps. However, it turns out that the gaps of one operator overlap with the spectral bands of the other one, so that the full operator have no gap.

Proposition 10.

Proof.

Let us suppose that there exists such that (of course, the same is true for some open neighborhood of ). We first note that , since these sets are either in the spectrum of or in the spectrum (Propositions 5 and 9). As a consequence, and . Then, since , the characterizations (18)-(51) of the essential spectrum of and (divided respectively by and ) imply that

(53)

Introducing , the system (53) can be rewritten as
{numcases}
a24sin^2ω-asinωcosω+cos^2ω¿1,
14a2sin^2ω+1asinωcosω+cos^2ω¿1.
Multiplying (53) by and taking the sum with (53) we obtain

which is impossible.
∎

Let us then remark that the set of eigenvalues of , which is the union of the sets of eigenvalues of and , is embedded in the essential spectrum of .

5.1 Main result

We return now to the case of the ladder. As it was mentioned before,
instead of studying the full operator we will
study separately the operators ,
.
Let us remind first the
result, already proven for instance in
[39],
which states the convergence of the essential spectrum of the periodic
operators (resp. ) to the essential
spectrum of (resp. ). The proof in [39] is based on the convergence of the eigenvalues of the reduced operator . We point out that the construction of the asymptotic expansion of these eigenvalues in the vicinity of the intersection point of the dispersion curves of is delicate and we refer the reader to [43] for an example of detailed asymptotic in that case

.

Theorem 3 (Essential spectrum).

Let {(am,bm),m∈N∗} be the gaps of the operator As (respectively Aa) on the limit graph G. Then, for each m0∈N∗ there exists ε0>0 such that if ε<ε0 the operator Aε,s (respectively Aε,a) has at least m0 gaps {(aε,m,bε,m),1⩽m⩽m0} such that

aε,m=am+O(ε),bε,m=bm+O(ε),ε→0,1⩽m⩽m0.

In [51], O. Post proves the norm convergence of the resolvent
of the laplacian with Neumann boundary conditions for a
large class of thin domains shrinking to graphs. It
consequently demonstrates the existence of eigenvalues of
Aμε,s (respectively Aμε,a) located in the gap of the essential
spectrum. This paper provides a simple and constructive alternative proof of this result. In the following proof, we consider the eigenvalues of the operator Aμε,s, the case of the operator Aμε,a being treated analogously.

Theorem 4 (Discrete spectrum).

Let (a,b) be a gap of the operator Aμs (respectively Aμa) on the limit graph G and λ∈(a,b) an eigenvalue of this operator. Then there exists ε0>0 such that if ε<ε0 the operator Aμε,s (resp. Aμε,a) has an eigenvalue λε inside a gap (aε,bε). Moreover, for ε<ε0, there exists C>0 such that

|λε−λ|≤C√ε.

(54)

Remark 6.

As every eigenvalue of the operators Aμs (resp. Aμa) is simple (as established in Theorem 1), for ε small enough, λε will be a simple eigenvalue of Aμε,s (resp. Aμε,a), see [51]. In addition, it is worth noting that the error estimate (54) is suboptimal. In fact, writing a high order asymptotic expansion of λε restores the optimal convergence rate:

|λε−λ(0)|≤Cε

The construction and the justification of this high order asymptotic expansion will be detailed in a forthcoming paper.

Remark 7.

We point out that imposing Dirichlet conditions leads to an entirely different asymptotic analysis. In [24], the asymptotic of the eigenvalues is obtained in the case of a compact ’thickened’ graph with different types of boundary conditions, including the Dirichlet and Robin ones (see also [5] for non standard boundary conditions). Besides, the Dirichlet ladder is investigated in [40]-[45]: as in our case, changing the size of one or several rungs of the ladder can create eigenvalues inside the first gap ( see [45, Theorem 8.1]). The analysis is deeply linked to the presence of a non empty discrete spectrum for the Laplace problem posed in a T-shape waveguide (cf. [46]).

Thanks to Theorem 4, it is easy to show the existence of as many eigenvalues as one wants.

Corollary 1.

For any number m∈N, it exists ε0 such that for all ε≤ε0, Aμε,s (respectively Aμε,a) has at least m eigenvalues.

In the next section, we give the constructive proof and in Section 5.3, we illustrate these theoretical results by numerical illustrations.

Our proof of Theorem 4 relies on the construction of a pseudo-mode,
that is to say a symmetric function uε∈H1(Ωμε) such that for every symmetric function v∈H1(Ωμε)

∣∣∫Ωμε(∇uε∇v−λuεv)dx∣∣⩽C√ε∥uε∥H1(Ωμε)∥v∥H1(Ωμε),

(55)

By adapting the Lemma 4 for [43] (see Appendix A in [11]) the existence of such a function provides an estimate of the
distance from λ to the spectrum of Aμε,s, namely

dist(σ(Aμε,s),λ)⩽˜C√ε,

(56)

with some constant ˜C that does not depend on
ε, but depends on λ.

According to Theorem 3, for ε
small enough, there exists a constant C such that
σess(Aμε,s)∩[a+Cε,b−Cε]=∅. As a consequence, the
intersection between the discrete spectrum
σd(Aμε,s) and the interval [λ−˜C√ε,λ+˜C√ε]
is non empty, which proves the existence of an eigenvalue in the neighborhood of λ.

Construction of a pseudo-mode

Since we consider the symmetric case, it suffices to construct the pseudo-mode uε
on the lower half part Ωμ,−ε
of Ωμε (comb shape domain, see Figure 8):

Ωμ,−ε={(x,y)∈Ωμε \;s.t. y<0}.

Figure 8: The domain Ωμ,−ε

As represented on Figure 8, we denote by Eε,−j+12, j∈Z, the horizontal
edges of the domain Ωμ,−ε

Eε,−j+12=(j+εμj/2,(j+1)−εμj+1/2)×(−L/2,−L/2+ε),

by Eε,−j, j∈Z, its
vertical edges

Eε,−j=(j−εμj/2,j+εμj/2)×(−L/2+ε,0),

and by Kε,−j, j∈Z, the junctions

Kε,−j=(j−εμj/2,j+εμj/2)×(−L/2,−L/2+ε).

Here, μj=1 if j≠0 and μ0=μ (with the notations of Section 4.1μj=wμ(ej), the function wμ being defined by (9)).
Denoting by u
an eigenfunction of the limite operator Aμs associated with the
eigenvalue λ (see formula (36)), we construct the pseudo-mode uε on Ωμ,−ε by ”fattening” u (with an appropriate rescaling) as follows:

The pseudo-mode being constructed, it remains to prove (55). We notice that it is sufficient to prove it for any test function
v∈C1s(¯¯¯¯¯¯¯Ωμε) (C1s standing
for the symmetric subspace of C1). Indeed,
C1s(¯¯¯¯¯¯¯Ωμε) is dense in the subset of H1(Ωμε) made of symmetric functions. Let us then estimate
the left-hand side of (55) for v∈C1s(¯¯¯¯¯¯¯Ωμε).
First, an integration by parts gives

Since uj is
a geometrical progression (according
to (44)), and using the Cauchy Schwarz inequality (the size of the junction
Kε,−j is of
order ε2), we obtain

∑j∈Zλuj∣∣∫Kε,−jvdx∣∣⩽C1ε∥v∥L2(Ωμ,−ε).

(61)

where C1 is a constant that does not depend on ε.
Next, denoting by Mε,−j the barycenter of Kε,−j, we remark that we can replace v(x,y) by v(x,y)−v(Mε,−j) in the integrals over the boundaries in the right-hand side of (60) because u satisfies the Kirchhoff’s
conditions (13). Moreover,

∣∣∫Γεj(v(x,y)−v(Mε,−j))dx∣∣⩽∫Γεj(x,y)∫Mε,−j|∇v|dxdt⩽C2ε∥v∥H1(Kε,−j).

(62)

In the previous formula, (x,y)∫Mε,−j stands for the integral on the segment linking Mε,−j to the point of coordinates (x,y).
Combining (60-61-62) and
taking into account (35-36-44), we
obtain that

∣∣∫Ωμ,−ε(∇uε∇v−λuεv)dx∣∣⩽C3ε∥v∥H1(Ωμ,−ε),∀v∈C1s(Ωμε).

(63)

To conclude, we notice that by definition of the pseudo-mode uε,

∥uε∥H1(Ωμ,−ε)⩾C4√ε∥u∥H1(G−),C4>0,

(G− standing for the lower half part of the graph G), which, together with (63) and the density argument mentioned above finishes the proof of (55).

5.3 Numerical illustration

To illustrate and validate the results of Theorem 4, we have computed a part of the essential spectrum, some eigenvalues and their associated eigenvectors of the operator Aμε,s for several values of ε and several values of μ.

Essential spectrum

To compute the essential spectrum of the operator Aμε,s (resp. As), one method consists in computing the eigenvalues λ(n)ε(θ)=(ω(n)ε(θ))2 defined in (6) (resp. λ(n)s(θ)=(ω(n)s(θ))2) defined in (17)) for a discrete set of θ included in [−π,π[ (or equivalently in [−π,0]). From a numerical point of view, this is done using the standard P1 conform finite element method ([23]). We have represented in Figure 9 the dispersive curves θ↦ω(n)ε(θ) for n∈\llbracket1,5\rrbracket, when L=2 and ε=0.1 (left figure), and θ↦ω(n)s(θ) for n∈\llbracket1,5\rrbracket, corresponding to the graph with the same L (right figure).

Figure 9: Dispersive curves for the ladder of thickness ε=0.1 (left figure) and for the graph (right figure) when L=2.

The essential spectrum can be easily deduced from the dispersion curves: indeed, as explained by the Floquet-Bloch theory, it is the image of the segment ([−π,π]) by the functions λ(n)ε (resp. λ(n)s). In Figure 10, we have represented a part of the essential spectrum of the operator Aμε,s for different values of ε: the blue bars correspond to the values ω such that λ=ω2 is in the essential spectrum of Aμε,s.

Figure 10: Representation of the essential spectrum of Aμε,s for different values of ε for L=2

Obviously, for small values of ε, the essential spectrum of the operator Aμε,s is very close to the essential spectrum of the limit operator Aμs. More precisely, to each gap of the limit operator Aμs, corresponds a gap of the operator Aμε,s, which is close to it for small ε . The convergence with respect to ε is linear as it is predicted by the theory, see Theorem 3. We also remark a phenomenon that has not been detected by our approach: the opening of a gap near the values ω=πN∗, points where the dispersion curves of the limit operator Aμs touch (see right figure of Figure 9). This phenomenon could be probably proven using the techniques of [43].

Another interesting phenomenon concerns the eigenvalues of infinite multiplicity for the limit operator. As explained in Remark 3, the operator Aμs might have eigenvalues of infinite multiplicity when L is rational. For instance in the case of L=0.5, the set of eigenvalues of infinite multiplicity is given by

{λ=ω2,ω=2(2n+1)π,n∈N}.

We can predict that such an eigenvalue becomes a (small) spectral band in the 2D case for ε small enough. Indeed, as shown in [51], the dimension of the spectral projector on any interval is preserved for ε small enough. On the other hand, in most cases, a periodic 2D operator does not have eigenvalues (see for instance [56, 57, 22] for the proof of the absolute continuity of the spectrum of classes of periodic operator defined in waveguides, but see also the counterexample [19]). Thus the most likely possibility is that the operator Aμε,s has a small spectral band (of width O(ε), see Theorem 3) in a neighborhood of the eigenvalues of infinite multiplicity. This phenomenon can be seen on Figure 11 where the essential spectrum of Aμε,s is represented for different values of ε and for L=0.5. A small spectral band appears in the vicinity of ω=2π, corresponding to the first eigenvalue of infinite multiplicity of the limit operator.

Figure 11: Representation of the essential spectrum of Aμε,s for different values of ε for L=0.5

Discrete spectrum

It is less easy to compute the discrete spectrum because one has to solve an eigenvalue problem set on an unbounded domain. To address this difficulty, we have used a method based on the construction of Dirichlet-to-Neumann operators in periodic waveguides (see [28, 20]): this requires the solution of cell problems (discretized here again using the standard P1 finite element methods) and the solution of a stationary Ricatti equation. The construction of these Dirichlet-to-Neumann operators enables us to reduce the numerical computation to a small neighborhood of the perturbation independently from the confinement of the mode (which depends on the distance between the eigenvalue and the essential spectrum of the operator). However the reduction of the problem leads to a non linear eigenvalue problem (since the DtN operators depend on the eigenvalue) of a fixed point nature. It is solved using a Newton-type procedure, each iteration needing a finite element computation, see [21] for more details.

Figure 12: Representation of the eigenvalues appearing in the first gap of the operator Aμε,s for different values of ε for L=2 and μ=0.25 (red asterisks). The values for ε=0 correspond to the limit operator Aμs.

In Figure 12, we represent the eigenvalues computed for different values of ε: here again, the blue bars correspond to the values ω such that λ=ω2 is in the essential spectrum of Aμε,s. The red asterisks stand for the values ω such that λ=ω2 is in the discrete spectrum of Aμε,s. In Figure 13, we make a zoom on the eigenvalues and we observe the linear convergence of one of this eigenvalue toward the limit one: as explained in Remark 6, the error estimate (54) is suboptimal. Indeed, a high order asymptotic expansion of λε would restore the linear convergence rate. In Figure 14, the eigenfunction corresponding to the first eigenvalue of the operator Aμε,s is represented.

Figure 13: Linear convergence of the eigenvalues represented in figure 12 as ε→0.Figure 14: Eigenfunction corresponding to the first eigenvalue of the operator Aμε,s for L=2, ε=0.06 and μ=0.25.

In Figure 15, we study the dependence of the eigenvalues with respect to μ∈(0,1). As it is natural to expect, the smaller μ is (so, the stronger the perturbation is), the better the eigenvalues are separated from the essential spectrum. When μ is close to 1, the computation becomes more costly: since the distance between the eigenvalue and the essential spectrum becomes very small, the mesh size has to be small enough in order to make the distinction between the two different kinds of spectrum.

Figure 15: Dependence of the eigenvalues in the first gap with respect to μ for L=2 and ε=0.1.

A last natural question for which no theoretical answer has been given yet is what happens for larger values of ε, i.e. when the spectrum of the operator Aμε,s is not close to the spectrum of the limit operator. In particular, if a gap exists for small values of ε, does it still exist for large values of ε (until the obstacles disappear)? Similarly, do the eigenvalues still exist when ε increases or do they immerse into the essential spectrum?
In the cases that we have tested, the gaps seem to keep present for any value of ε for which the obstacles are present (for ε∈(0,min(1,L/2))). In Figure 16, we represent the dependence of the first two gaps with respect to ε in the case L=2. The limit case ε→min(1,L/2) has been studied by S. Nazarov [43] where opening of a gap is proven when Dirichlet conditions are imposed on the boundary of the periodic waveguide instead of Neumann boundary conditions. The behaviour of the eigenvalues is by contrast more unclear. In Figure 17, we show the eigenvalues in the first gap of the operator Aμε,s for L=2 and μ=0.25. We might think that the eigenvalues disappear for some values of ε<1: however, as previously mentioned, the numerical computation becomes costly when the eigenvalues approach the essential spectrum. For this reason, it is difficult to make the distinction between the case when the eigenvalues do not exist any more and the case when they exist but are very close to the essential spectrum. Moreover, if they really disappear, do they immerse in the essential spectrum? Do they move in the complex plane? Let us point out that the use of the sophisticated numerical method (based on an automatic choice of the mesh size) presented in [31] might help to answer these questions.

Figure 16: Dependence of the first gaps with respect to ε<1 for L=2.Figure 17: Dependence of the eigenvalues in the first gap with respect to ε<1 for L=2 and μ=0.25.

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